Download 3.2 Solve Linear Systems Algebraically

3.2
Before
Solve Linear Systems
Algebraically
You solved linear systems graphically.
Now
You will solve linear systems algebraically.
Why?
So you can model guitar sales, as in Ex. 55.
Key Vocabulary
In this lesson, you will study two algebraic methods for solving linear systems.
• substitution method The first method is called the substitution method.
• elimination method
For Your Notebook
KEY CONCEPT
The Substitution Method
STEP 1
Solve one of the equations for one of its variables.
STEP 2 Substitute the expression from Step 1 into the other equation
and solve for the other variable.
STEP 3 Substitute the value from Step 2 into the revised equation from
Step 1 and solve.
EXAMPLE 1
Use the substitution method
Solve the system using the substitution method.
2x 1 5y 5 25
x 1 3y 5 3
Equation 1
Equation 2
Solution
STEP 1
Solve Equation 2 for x.
x 5 23y 1 3
Revised Equation 2
STEP 2 Substitute the expression for x into Equation 1 and solve for y.
2x 1 5y 5 25
2(23y 1 3) 1 5y 5 25
y 5 11
Write Equation 1.
Substitute 23y 1 3 for x.
Solve for y.
STEP 3 Substitute the value of y into revised Equation 2 and solve for x.
x 5 23y 1 3
Write revised Equation 2.
x 5 23(11) 1 3
Substitute 11 for y.
x 5 230
Simplify.
c The solution is (230, 11).
CHECK Check the solution by substituting into the original equations.
2(230) 1 5(11) 0 25
25 5 25 ✓
160
n2pe-0302.indd 160
Substitute for x and y.
Solution checks.
230 1 3(11) 0 3
353✓
Chapter 3 Linear Systems and Matrices
10/20/05 3:39:57 PM
ELIMINATION METHOD Another algebraic method that you can use to solve a
system of equations is the elimination method. The goal of this method is to
eliminate one of the variables by adding equations.
For Your Notebook
KEY CONCEPT
The Elimination Method
STEP 1 Multiply one or both of the equations by a constant to obtain
coefficients that differ only in sign for one of the variables.
STEP 2 Add the revised equations from Step 1. Combining like terms will
eliminate one of the variables. Solve for the remaining variable.
STEP 3 Substitute the value obtained in Step 2 into either of the original
equations and solve for the other variable.
EXAMPLE 2
Use the elimination method
Solve the system using the elimination method.
3x 2 7y 5 10
6x 2 8y 5 8
Equation 1
Equation 2
Solution
STEP 1
Multiply Equation 1 by 22 so that the coefficients of x differ only in sign.
3x 2 7y 5 10
SOLVE SYSTEMS
In Example 2, one
coefficient of x is a
multiple of the other. In
this case, it is easier to
eliminate the x-terms
because you need
to multiply only one
equation by a constant.
26x 1 14y 5 220
3 22
6x 2 8y 5 8
6x 2 8y 5 8
STEP 2 Add the revised equations and solve for y.
6y 5 212
y 5 22
STEP 3 Substitute the value of y into one of the original equations. Solve for x.
3x 2 7y 5 10
Write Equation 1.
3x 2 7(22) 5 10
Substitute 22 for y.
3x 1 14 5 10
Simplify.
4
x 5 2}
Solve for x.
3
4 , 22 .
c The solution is 1 2}
2
3
CHECK You can check the solution algebraically
using the method shown in Example 1. You can
also use a graphing calculator to check the solution.
"MHFCSB
✓
GUIDED PRACTICE
Intersection
X=-1.333333 Y=-2
at classzone.com
for Examples 1 and 2
Solve the system using the substitution or the elimination method.
1. 4x 1 3y 5 22
x 1 5y 5 29
2. 3x 1 3y 5 215
5x 2 9y 5 3
3. 3x 2 6y 5 9
24x 1 7y 5 216
3.2 Solve Linear Systems Algebraically
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★
EXAMPLE 3
Standardized Test Practice
To raise money for new football uniforms, your school sells silk-screened
T-shirts. Short sleeve T-shirts cost the school $5 each and are sold for $8
each. Long sleeve T-shirts cost the school $7 each and are sold for $12 each.
The school spends a total of $2500 on T-shirts and sells all of them for
$4200. How many of the short sleeve T-shirts are sold?
A 50
B 100
C 150
D 250
Solution
STEP 1
Write verbal models for this situation.
Equation 1
Short sleeve
cost
Short sleeve
shirts
p
(dollars/shirt)
5
1
(shirts)
Long sleeve
cost
p
Long sleeve
shirts
(dollars/shirt)
5
(shirts)
Total
cost
(dollars)
p
x
1
7
p
y
5
2500
p
Short sleeve
shirts
1
Long sleeve
selling price
p
Long sleeve
shirts
5
Total
revenue
Equation 2
Short sleeve
selling price
(dollars/shirt)
8
(shirts)
p
x
(dollars/shirt)
p
12
1
(shirts)
y
(dollars)
5
4200
STEP 2 Write a system of equations.
Equation 1
Equation 2
5x 1 7y 5 2500
8x 1 12y 5 4200
Total cost for all T-shirts
Total revenue from all T-shirts sold
STEP 3 Solve the system using the elimination method.
Multiply Equation 1 by 28 and Equation 2 by 5 so that the
coefficients of x differ only in sign.
5x 1 7y 5 2500
8x 1 12y 5 4200
3 28
35
Add the revised equations and solve for y.
AVOID ERRORS
Choice D gives the
number of long sleeve
T-shirts, but the
question asks for the
number of short sleeve
T-shirts. So you still
need to solve for x in
Step 3.
240x 2 56y 5 220,000
40x 1 60y 5 21,000
4y 5 1000
y 5 250
Substitute the value of y into one of the original equations
and solve for x.
5x 1 7y 5 2500
5x 1 7(250) 5 2500
5x 1 1750 5 2500
x 5 150
Write Equation 1.
Substitute 250 for y.
Simplify.
Solve for x.
The school sold 150 short sleeve T-shirts and 250 long sleeve T-shirts.
c The correct answer is C. A B C D
162
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Chapter 3 Linear Systems and Matrices
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✓
GUIDED PRACTICE
for Example 3
4. WHAT IF? In Example 3, suppose the school spends a total of $3715 on
T-shirts and sells all of them for $6160. How many of each type of T-shirt
are sold?
CHOOSING A METHOD In general, the substitution method is convenient when
one of the variables in a system of equations has a coefficient of 1 or 21, as in
Example 1. If neither variable in a system has a coefficient of 1 or 21, it is usually
easier to use the elimination method, as in Examples 2 and 3.
EXAMPLE 4
Solve linear systems with many or no solutions
Solve the linear system.
a. x 2 2y 5 4
b. 4x 2 10y 5 8
3x 2 6y 5 8
214x 1 35y 5 228
Solution
a. Because the coefficient of x in the first equation is 1, use the
substitution method.
Solve the first equation for x.
x 2 2y 5 4
Write first equation.
x 5 2y 1 4
Solve for x.
Substitute the expression for x into the second equation.
3x 2 6y 5 8
Write second equation.
3(2y 1 4) 2 6y 5 8
Substitute 2y 1 4 for x.
12 5 8
Simplify.
c Because the statement 12 5 8 is never true, there is no solution.
b. Because no coefficient is 1 or 21, use the elimination method.
AVOID ERRORS
When multiplying
an equation by a
constant, make sure
you multiply each term
of the equation by the
constant.
✓
Multiply the first equation by 7 and the second equation by 2.
4x 2 10y 5 8
37
28x 2 70y 5 56
214x 1 35y 5 228
32
228x 1 70y 5 256
Add the revised equations.
050
c Because the equation 0 5 0 is always true, there are infinitely many
solutions.
GUIDED PRACTICE
for Example 4
Solve the linear system using any algebraic method.
5. 12x 2 3y 5 29
6. 6x 1 15y 5 212
24x 1 y 5 3
22x 2 5y 5 9
8. 12x 2 2y 5 21
3x 1 12y 5 24
9. 8x 1 9y 5 15
5x 2 2y 5 17
7. 5x 1 3y 5 20
3 y 5 24
2x 2 }
5
10. 5x 1 5y 5 5
5x 1 3y 5 4.2
3.2 Solve Linear Systems Algebraically
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3.2
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 5, 29, and 59
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 40, 50, 57, 58, and 60
SKILL PRACTICE
1. VOCABULARY Copy and complete: To solve a linear system where one of the
coefficients is 1 or 21, it is usually easiest to use the ? method.
2. ★ WRITING Explain how to use the elimination method to solve a linear
system.
EXAMPLES
1 and 4
on pp. 160–163
for Exs. 3–14
SUBSTITUTION METHOD Solve the system using the substitution method.
3. 2x 1 5y 5 7
4. 3x 1 y 5 16
x 1 4y 5 2
6. x 1 4y 5 1
9. 3x 1 2y 5 6
8. 3x 2 4y 5 25
6x 1 3y 5 14
2x 1 3y 5 25
10. 6x 2 3y 5 15
x 2 4y 5 212
11. 3x 1 y 5 21
22x 1 y 5 25
12. 2x 2 y 5 1
2x 1 3y 5 18
13. 3x 1 7y 5 13
8x 1 4y 5 6
on pp. 161–163
for Exs. 15–27
23x 1 y 5 7
7. 3x 2 y 5 2
3x 1 2y 5 212
EXAMPLES
2 and 4
5. 6x 2 2y 5 5
2x 2 3y 5 24
14. 2x 1 5y 5 10
x 1 3y 5 27
23x 1 y 5 36
ELIMINATION METHOD Solve the system using the elimination method.
15. 2x 1 6y 5 17
16. 4x 2 2y 5 216
17. 3x 2 4y 5 210
2x 2 10y 5 9
23x 1 4y 5 12
6x 1 3y 5 242
18. 4x 2 3y 5 10
19. 5x 2 3y 5 23
8x 2 6y 5 20
20. 10x 2 2y 5 16
2x 1 6y 5 0
21. 2x 1 5y 5 14
5x 1 3y 5 212
22. 7x 1 2y 5 11
3x 2 2y 5 236
24. 2x 1 5y 5 13
6x 1 2y 5 213
23. 3x 1 4y 5 18
22x 1 3y 5 29
6x 1 8y 5 18
25. 4x 2 5y 5 13
26. 6x 2 4y 5 14
6x 1 2y 5 48
2x 1 8y 5 21
27. ERROR ANALYSIS Describe and correct
the error in the first step of solving the
system.
3x 1 2y 5 7
5x 1 4y 5 15
26x 2 4y 5 7
5x 1 4y 5 15
2x
5 22
x 5 222
CHOOSING A METHOD Solve the system using any algebraic method.
28. 3x 1 2y 5 11
4x 1 y 5 22
31. 4x 2 10y 5 18
22x 1 5y 5 29
34. 2x 1 3y 5 26
n2pe-0302.indd 164
24x 1 5y 5 210
32. 3x 2 y 5 22
5x 1 2y 5 15
35. 3x 1 y 5 15
3x 2 4y 5 25
2x 1 2y 5 219
37. 4x 2 y 5 210
38. 7x 1 5y 5 212
6x 1 2y 5 21
164
29. 2x 2 3y 5 8
3x 2 4y 5 1
30. 3x 1 7y 5 21
2x 1 3y 5 6
33. x 1 2y 5 28
3x 2 4y 5 224
36. 4x 2 3y 5 8
28x 1 6y 5 16
39. 2x 1 y 5 21
24x 1 6y 5 6
Chapter 3 Linear Systems and Matrices
10/20/05 3:40:03 PM
40. ★ MULTIPLE CHOICE What is the solution of the linear system?
3x 1 2y 5 4
6x 2 3y 5 227
A (22, 25)
B (22, 5)
C (2, 25)
D (2, 5)
GEOMETRY Find the coordinates of the point where the diagonals of the
quadrilateral intersect.
41.
42.
y
(1, 4)
y
(4, 4)
43.
(3, 7)
y
(5, 5)
(1, 3)
(7, 4)
(0, 2)
(1, 6)
(5, 0)
(6, 1)
x
(1, 21)
x
(7, 0)
x
SOLVING LINEAR SYSTEMS Solve the system using any algebraic method.
44. 0.02x 2 0.05y 5 20.38
0.03x 1 0.04y 5 1.04
2y 5 5
1x 1 }
47. }
}
3
6
2
5 x1 7 y5 3
}
}
}
4
12
12
45. 0.05x 2 0.03y 5 0.21
0.07x 1 0.02y 5 0.16
x13 1 y21 51
48. }
}
3
4
2x 2 y 5 12
2 x 1 3y 5 234
46. }
3
1 y 5 21
x2}
2
x21 1 y12 54
49. }
}
2
3
x 2 2y 5 5
50. ★ OPEN-ENDED MATH Write a system of linear equations that has (21, 4) as
its only solution. Verify that (21, 4) is a solution using either the substitution
method or the elimination method.
SOLVING NONLINEAR SYSTEMS Use the elimination method to solve the system.
51. 7y 1 18xy 5 30
13y 2 18xy 5 90
52. xy 2 x 5 14
53. 2xy 1 y 5 44
5 2 xy 5 2x
32 2 xy 5 3y
54. CHALLENGE Find values of r, s, and t that produce the indicated solution(s).
23x 2 5y 5 9
rx 1 sy 5 t
a. No solution
b. Infinitely many solutions
c. A solution of (2, 23)
PROBLEM SOLVING
EXAMPLE 3
on p. 162
for Exs. 55–59
55. GUITAR SALES In one week, a music store sold 9 guitars for a total of $3611.
Electric guitars sold for $479 each and acoustic guitars sold for $339 each.
How many of each type of guitar were sold?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
56. COUNTY FAIR An adult pass for a county fair costs $2 more than a children’s
pass. When 378 adult and 214 children’s passes were sold, the total revenue
was $2384. Find the cost of an adult pass.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
3.2 Solve Linear Systems Algebraically
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57. ★ SHORT RESPONSE A company produces gas mowers and electric
mowers at two factories. The company has orders for 2200 gas mowers and
1400 electric mowers. The production capacity of each factory (in mowers per
week) is shown in the table.
Factory A
Factory B
Gas mowers
200
400
Electric mowers
100
300
Describe how the company can fill its orders by operating the factories
simultaneously at full capacity. Write and solve a linear system to support
your answer.
58. ★ MULTIPLE CHOICE The cost of 11 gallons of regular gasoline and
16 gallons of premium gasoline is $58.55. Premium costs $.20 more per
gallon than regular. What is the cost of a gallon of premium gasoline?
A $2.05
B $2.25
C $2.29
D $2.55
59. TABLE TENNIS One evening, 76 people
gathered to play doubles and singles table
tennis. There were 26 games in progress at
one time. A doubles game requires 4 players
and a singles game requires 2 players. How
many games of each kind were in progress
at one time if all 76 people were playing?
60. ★ EXTENDED RESPONSE A local hospital is holding a two day marathon walk
to raise funds for a new research facility. The total distance of the marathon
is 26.2 miles. On the first day, Martha starts walking at 10:00 A.M. She walks
4 miles per hour. Carol starts two hours later than Martha but decides to run
to catch up to Martha. Carol runs at a speed of 6 miles per hour.
a. Write an equation to represent the distance Martha travels.
b. Write an equation to represent the distance Carol travels.
c. Solve the system of equations to find when Carol will catch up to Martha.
d. Carol wants to reduce the time she takes to catch up to Martha by 1 hour.
How can she do this by changing her starting time? How can she do this
by changing her speed? Explain whether your answers are reasonable.
61. BUSINESS A nut wholesaler sells a mix of peanuts and cashews. The
wholesaler charges $2.80 per pound for peanuts and $5.30 per pound
for cashews. The mix is to sell for $3.30 per pound. How many pounds
of peanuts and how many pounds of cashews should be used to make
100 pounds of the mix?
62. AVIATION Flying with the wind, a plane flew 1000 miles in 5 hours. Flying
against the wind, the plane could fly only 500 miles in the same amount of
time. Find the speed of the plane in calm air and the speed of the wind.
63. CHALLENGE For a recent job, an electrician earned $50 per hour, and the
electrician’s apprentice earned $20 per hour. The electrician worked 4 hours
more than the apprentice, and together they earned a total of $550. How
much money did each person earn?
166
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5 WORKED-OUT SOLUTIONS
Chapter 3 Linear
and Matrices
on p. Systems
WS1
★
5 STANDARDIZED
TEST PRACTICE
10/20/05 3:40:06 PM
MIXED REVIEW
Solve the equation.
64. 25x 1 4 5 29 (p. 18)
65. 6(2a 2 3) 5 230 (p. 18)
66. 1.2m 5 2.3m 2 2.2 (p. 18)
67. x 1 3 5 4 (p. 51)
68. 2x 1 11 5 3 (p. 51)
69. 2x 1 7 5 13 (p. 51)
Tell whether the lines are parallel, perpendicular, or neither. (p. 82)
70. Line 1: through (2, 10) and (1, 5)
71. Line 1: through (4, 5) and (9, 22)
Line 2: through (3, 27) and (8, 28)
Line 2: through (6, 26) and (22, 21)
Write an equation of the line. (p. 98)
72. y
73.
2
(2, 4)
74.
y
4
x
4
(5, 1)
1
y
(3, 1)
(22, 22)
(2, 4)
2
x
(21, 27)
1
x
PREVIEW
Graph the inequality in a coordinate plane. (p. 132)
Prepare for
Lesson 3.3
in Exs. 75–80.
75. x < 23
76. y ≥ 2
77. 2x 1 y > 1
78. y ≤ 2x 1 4
79. 4x 2 y ≥ 5
80. y < 23x 1 2
QUIZ for Lessons 3.1–3.2
Graph the linear system and estimate the solution. Then check the solution
algebraically. (p. 153)
1. 3x 1 y 5 11
x 2 2y 5 28
2. 2x 1 y 5 25
3. x 2 2y 5 22
2x 1 3y 5 6
3x 1 y 5 220
Solve the system. Then classify the system as consistent and independent,
consistent and dependent, or inconsistent. (p. 153)
4. 4x 1 8y 5 8
x 1 2y 5 6
5. 25x 1 3y 5 25
5x 1 1
y5}
3
6. x 2 2y 5 2
2x 2 y 5 25
Solve the system using the substitution method. (p. 160)
7. 3x 2 y 5 24
x 1 3y 5 228
8. x 1 5y 5 1
9. 6x 1 y 5 26
23x 1 4y 5 16
4x 1 3y 5 17
Solve the system using the elimination method. (p. 160)
10. 2x 2 3y 5 21
2x 1 3y 5 219
11. 3x 2 2y 5 10
12. 2x 1 3y 5 17
26x 1 4y 5 220
5x 1 8y 5 20
13. HOME ELECTRONICS To connect a VCR to a television set, you need a cable
with special connectors at both ends. Suppose you buy a 6 foot cable for
$15.50 and a 3 foot cable for $10.25. Assuming that the cost of a cable is the
sum of the cost of the two connectors and the cost of the cable itself, what
would you expect to pay for a 4 foot cable? Explain how you got your answer.
EXTRA PRACTICE for Lesson 3.2, p. 1012
n2pe-0302.indd 167
ONLINE
at classzone.com
3.2 Solve QUIZ
Linear Systems
Algebraically
167
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